A Trigger of Coupled Singularities

By using example of nonlinear dynamics of a pair of coupled gears, the phenomenon of appearance and disappearance of a trigger of coupled singularities and homoclinic orbits in the form of number eight in the phase portrait in the phase plane is investigated. That phenomenon is an accompanying phenomenon of loss of stability of the local unique equilibrium position. For a generalized case under certain conditions, a theorem of the appearance of a trigger of coupled singularities in a nonlinear dynamical conservative system, the first derivative of the system potential energy which is a product of two periodic functions with different periods, and one bifurcation parameter, which is the cause for the appearance of new roots of these two functions, is defined.